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Full text of "Treatise On Analysis Vol-Ii"

288       XIV   INTEGRATION IN LOCALLY COMPACT GROUPS

From the results of (14.9) we have, in particular:

(14.10.5) Suppose that f and g are locally p-integrable. If one of the functions
/, g is continuous, and if one has compact support, then f and g are convolvable,
the right-hand sides of (14.10.2) and (14.10.3) are defined for all x e G, and the
function f*g is continuous. If both f and g belong to Jfc(G), then so does
f*9-

This follows from (14.9.1) and (14.9.3).

(1 4.1 0.6)   Let f be a f$-integrable function .

(i)   For p = 1, 2 or +00, the function f is convolvable with every function
g e &G9 P); the function f * g belongs to J2?(G, 0); and

(14.10.6.1)

(ii)   Ifp = + oo, andg e S?g(G, j8), the integral

(xs-^ACs-1) dfts)

is defined for all xeG, and the function x\-+\f(xs~~1)g(s)A(s~'1)d{S(s) is
uniformly continuous with respect to every right-invariant distance on G.
(iii)   Ifp = 1, we have

(14.10.6.2)

(iv)   Ifg e *g(G), rten ^fao / * g e g(G).

Parts (i) and (iv) follow from (14.9.2) and the relation \\f-0\\ = N^/)
(13.20.3). To prove (14.10.6.2), we remark that by virtue of (14.8.1.1) and
the fact that the function (s9 x)\-+g(x)f(s) is (0  jS)-integrable (13.21.14),
the function (s,x)\r+g(s~lx)f(s) is also (p  jS)-integrable. The formula
(14.10.6.2) then comes immediately from the Lebesgue-Fubini theorem and
the left-invariance of p.

As to (ii), for each x e G the function sh-^Cs-"1*) belongs to 3?c(G), and
therefore the integral on the right-hand side of (14.10.2) is defined for all
x e G. If we put v = A"1  /?, then v is a right Haar measure (14.3.4), and we
may write (14.10.3) in the form

,,  f., _t

*)-j/(**e of bounded measures on G. Suppose that, for each function